ind an equation for the plane that passes through the point A(1,2,3)parallel to u = 2i + 3 j + k nd y =i- j+2k.

Q: Find the Cartesian equation of the plane containing the point P(1,3,-2) and the I T = (0, -1, 4) +…

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Q: Find the real number a such that the line of x = 1, y = 2-1, z=3+t, the plane of equation parametric…

A: Given parametric equations are x=t , y=2-t, z=3+t Where t∈ℝ and  Plane is αx+5y+z-10=0

Q: Find the trace of the plane in the given coordinate plane.3x − 9y + 4z = 5,  yz

A: The trace of the plane in yz plane is obtained by substituting x=0 in the given equation of the…

Q: Consider a, B, t/= 0. A possible graphical representation for the plane of equation Bx + ay +t = 0…

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Q: Find the coordinates of the point where the line F = (1+ 2t)i + tj – 2tk intersects the plane 2x – y…

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Q: Find the vector equation for the line of intersection of the planes 5x +y-3z= -3 and 5x + z = -1 r =…

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Q: Let r(t) =(t, 4/3 t3/2, t2) Find the equation of the osculating plane at the point correspondingto t…

A: Given that, rt=t, 43t32, t2rt=t, 2t33, t2

Q: (b) Suppose that the position vector of a particle moving in the plane is r-12vii+j, t>0. Find the…

A: This is application of derivative problem

Q: Let y = x² be a parabola in the xy-plane parametrized by r(t) = (t,t², 0). What are the vectors T, N…

A: Solution is given below:

Q: Find equations for the planes The plane through P0(0, 2, -1) normal to n = 3i - 2j - k

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Q: Is v = 2i - 4j + k related in any special way to the plane 2x + y = 5? Give reasons for your answer.

A: Given:

Q: Find the trace of the plane in the given coordinate plane.3x − 9y + 4z = 5,  xz

A: Let the plane P whose equation is:                                          3x-9y+4z=5…

Q: Find a vector parametric equation for the ellipse that lies on the plane 5y−3x+z=9 and is inside the…

A: Given that: 5y-3x+z=9 Let x=u cos v, y=u sin v So z=9+3x-5y  =9+3u cos v-5u sin v

Q: ind the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1,2), C = (3,…

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Q: M (1; -2; 4) parallel to the OXZ plane

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Q: a) Evaluate ff, (2x – z) dS, where S is part of the plane 3x + 2y +z = 1 that lies above the…

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Q: Where does the parametric line ( 2 + 2 t , t , t ) intersect the plane x+y-3z=4?

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Q: Find the real number a such that the line of parametric equations x = t, y = 2 – t, z = 3 + t, te R…

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Q: Show that the curve a(t) = (t,4,²) lies in a plane and find the equation of this plane.

A: Solution:- αt=t,1+tt,1-t2t t,1t+1,1t-t=x,y,z It is clear that, t-1t-1+1t-t=x-y+z x-y+z=-1

Q: 1. Find an equation for the plane through P,(0,2,–1) perpendicular to n = 3i – 2j-k

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: The line X=t, y+1=t, Z=t intersects the plane. x-2y+Z=2 at the point

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Q: Find constants a, b, c such that  G( u, v) = ( u + av, bu + v, 2u - c) parametrizes the plane 3x - 4…

A: Given G( u, v) = ( u + av, bu + v, 2u - c) parametrizes the plane 3x - 4 y + z = 5.

Q: The equations for the plane through p0(0,2,-1) normal to n =3i - 2j - k O 3x-3y-3z=0 O 3x-3y=z O…

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Q: Find the vector equation that represents the curve of intersection of the paraboloid z 2x2 + y and…

A: z=2x2+y2  & x=ey We can see that y can be represented in terms of x.

Q: 12. a. Use vectors to show that the distance from P;(x1, Y1, Z1) to the plane Ax + By + Cz = D is…

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Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? A) (2 +…

A: The direction ratio of the normal vector to the plane is (3, 2,1 ) So the perpendicular line to the…

Q: Show that the curve a(t) = (t, , ) lies in a plane and find the equation of this plane.

A: For all 't' , we have α'(t) = 1 , -1t2 ,-1 - 1t2α"(t) = 0 , 2t3 , 2t3 Now  α'(t) × α"(t) = 2t3 ,…

Q: Let R be the triangle in the xy-plane with vertices (0,0),(0, 1),(1,0). Find C[n(2 – x – y) dA.

A: Given vertices of triangle now we have to find the area as shown below

Q: Find the general equation of the plane containing the point (2, –1,4) and normal to the line 2 = =…

A: As the line is normal to the plane, the DRs of the line and normal of plane is same.

Q: Find an equation for the plane passing through the origin that is parallel to the tan- gent plane to…

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Q: Let r(t)=(t, (4/3)t^(3/2), t^2). Find the equation of the osculating plane at the point…

A: Let the given point be (x0, y0, z0), to find the osculating plane find the cross product of the…

Q: Find a parametrization r(t,s) for the plane which contains the point P(-9, 7, 1) and the line…

A: Use plane concept

Q: Find an equation for the plane that passes through the point (1, 2, 3) parallel to u = 2i + 3j + k…

A: Given:u = 2i + 3j + k and v = i - j + 2kFirst find cross product

Q: Compute the parametric equations of the line in R3 determined by the intersection of the two planes:…

A: To obtain the parametric equations of the intersection line, use the following formulae. Let n1→…

Q: Find the equation of the plane passing through the three points A = (-1,0,2), B= (2, –1,2), C = (3,…

A: Given problem:- Find the equation of the plane passing through the three points A = (- 1, 0, 2) B =…

Q: 1. Show that A(r - ro) + B(y - yo) =0 is an equation for the line in the ry-plane through the point…

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Q: a) Show that the line I = 0, y = t, z =t where te R a) lies in the plane 6z+4y–4x = 0 b) is parallel…

A: In the question it is asked to solve for the planes having equation given below.

Q: Find the general form of the equation of the plane passing through the point and normal to the…

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Q: Find the equation of the plane passing through the three points A = (-1,0,2), B = (2,–1,2), C = (3,…

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Q: 4) Find equations for the planes: a. The plane through P(0,2,-1) normal to n = 3i-2j-k

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Q: Find an equation of the plane through the origin that is parallel to the plane 4x – 3y + 6z + 18 =…

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Q: Find the vector equation that represents the curve of intersection of the paraboloid z = 4x2 + y and…

A: The given curves are: z=4x2+y2     and     x=ey

Q: Find the vector equation for the line of intersection of the planes x – 5y + 2z = 1 and æ + z = 1 |…

A: To find the vector equation of the line of intersection of given planes.

Q: Find the point on the plane x+y+z%3D24 that is closest to the point (1,1,1).

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Q: Find the real number a such that the line of x = 1, y = 2 – 1, z = 3 +t, parametric equations te R…

A: Given, parametric equations are                               x = t, y = 2-t, z = 3 +t, t in R To…

Q: Let h VI and y: 8 For what value(s) of h, is y in the plane generated by vị and v2?

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Q: Find the parametric equations for the line of the intersection L of the two planes. 6x – 3y+ z = 5…

A: Parametric equations of the line is :  x(t) = x0 + a t  y(t) = y0 + b t  z(t) = z0 + c t  Here ( x0…

Q: (a) Find a vector parametric equation for the ellipse that lies on the plane 3x + 2y + z = -4 and…

A:

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.